MATH 1630 ~ Chapter 1
Solving Systems of Linear Equations
To solve a linear system of two equations means to find all ordered pairs of real numbers which satisfy
both equations.
Using Substitution
− 4 = 2x − y

Solve: 
 x − 2y = 2
a) Solve one of the equations for one of the variables
(Choose the equation which is easier to solve)
x = 2y + 2
b) Substitute this quantity for “x” in the other equation
-4 = 2(2y + 2) - y
c) Solve for “y”
-4 = 4y + 4 - y
-8 = 3y
8
y=−
3
16
10
 8
x = 2−  + 2 = −
+2= −
3
3
 3
d) Use back substitution to find “x”
e) Check your results by substituting into both original equations
 10   8 
−4 = 2  −  −  − 
 3   3
20 8
12
−4 = −
+ =−
= −4
3
3
3
f)
T
10
 8
− 2−  = 2
3
 3
10 16
6
−
+
=2⇒2=
3
3
3
−
8
 10
The solution is  − , − 
3
 3
Using Elimination
− 4 = 2x − y

Solve: 
 x − 2y = 2
 2x − y = −4

 x − 2y = 2
a) Write both equations in standard form
b) Use multiplication to create opposite
coefficients on one of the variables
M(-2)
 − 4x + 2y = 8

x − 2y = 2

Solving Systems of Linear Equations ~ p. 1
J. Ahrens ~ 2006
T
c) Add like terms and solve for “x”
−4x + 2y = 8
x − 2y = 2
−3x
d) Back substitute to find “y”
−
= 10
10
x=−
3
10
− 2y = 2
3
10
−2y =
+2
3
16
−2y =
3
8
y=−
3
e) Check as above
f)
A system with a single solution is consistent (and independent)
Graphically
[-10, 10] x [-10, 10]
Consistent: single solution
Convert answers to fractions.
(NOTE: Occasionally the calculator cannot convert a rational answer to a decimal due to a
round off error.)
Solving Systems of Linear Equations ~ p. 2
J. Ahrens ~ 2006
An Inconsistent System
−2 = x+y

Solve: 
 − 2x − 2y = 2
2x + 2y = −4
−2x − 2y = 2
Solve, using any analytic method
0 = −2
Unless we have made a mistake, this system has no solution.
A system with no solution is inconsistent.
The graph of the system is a pair of parallel lines.
Inconsistent system: no solution
It is impossible to guarantee these lines are parallel from the calculator display. The graph is only
used as a check.
A Consistent System with Multiple Solutions
−1= x + y

Solve: 
 − 2x − 2y = 2
2x + 2y = −2
Solve, using any analytic method
−2x − 2y = 2
0= 0
Unless we have made a mistake, there is something
unusual going on here. When we graph the system, we
find that the two lines coincide. (Turn the “bubble” on
for Y2 to confirm this assertion.)
This system is (consistent and) dependent. It has an
infinite number of solutions.
Since the lines are the same, any ordered pair which satisfies one equation will satisfy both.
The solution set is written as: {(x, y) | x + y = –1}
The solution set is read as: “The set of all ordered pairs (x, y) such that x + y = –1.”
Dependent system: multiple solutions
Solving Systems of Linear Equations ~ p. 3
J. Ahrens ~ 2006